Suppose I have the piecewise function $f(x) = \begin{cases}1, & 0<x<1 \\ 0 & x>1 \end{cases}$

and I wanted to extend it to an even function over the entire real line. How would I do that?

I know that normally for a function $\phi$ defined on the half line, the even extension would normally be $\phi_{\text{even}}(x)=\begin{cases} \phi(x)& x\geq 0 \\ +\phi(-x)& x\leq 0\end{cases}$, but how do I take care of a case when the function is constant on each piece, and thus has no "$x$'s" to substitute in a negative for?

I don't have a prof to ask - I'm trying to learn this on my own, so please don't be snarky if I sound clueless!

Thank you ahead of time.

  • $\begingroup$ ALannister: please note that the tag even-and-odd-extensions is currently up for discussion on Meta. I believe you may have been the user who created this tag. $\endgroup$ – Clarinetist Jan 18 '18 at 14:23

The extension is not unique since it is not known that what are $f(0)$ and $f(1)$. On the other hand, all you need is defined for $x<0$, $$ f(x):=f(-x). $$

  • $\begingroup$ so it would be $1$ on $(-1,0)$ and $0$ for $x<-1$? $\endgroup$ – ALannister Jan 7 '17 at 2:35
  • $\begingroup$ Yes. ${}{}{}{}{}{}{}{}{}{}{}$ $\endgroup$ – Jack Jan 7 '17 at 2:41

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